Grant is finally on the plane on his way to the Expo. He's curious about how far his plane is traveling though. He's not super familiar with the geography around the city where the Expo is, it's called something like Udacity, maybe its called Udacity. Hm, I don't know. However, he does know that Udacity is about 700 miles north of some other city that he knows a bit more about, and that city's only about 200 miles east of his town. To help him visualize the most efficient path for the plane to take, Grant wants to draw a picture, but a picture that's a bit more exact than the silly one I've drawn right here. If we say that Grant's hometown is located at the origin on some coordinate plane, and we split our coordinate plane up into chunks of square miles, what should the coordinates of Udacity be, and the coordinates of the other city be?
We know that this other city is directly east of Grant's hometown. So, we're not moving in the up and down or north and south direction at all when we go from here to here. We're only moving horizontally, which only changes the x-coordinate. We're moving over 200 miles so the x-coordinate becomes 200. Since you've moved 200 miles over from zero in the positive direction, and the y coordinate doesn't change at all. It stays as zero. We know that Udacity is directly north of this other city. So, if we move from the other city up to Udacity, all that we're changing is the y-coordinate, since we're only moving vertically. That means that the horizontal coordinate stays the same, but it needs to be the coordinate of this other city. So, Udacity is located at 200, and it's y-coordinate will be 0 plus 700, or 700.
Looking at this amateur map we've made, we can notice something actually pretty interesting,. These 3 cities on our map form a triangle. However, this isn't just any triangle. This is a right triangle, this angle right here is a
Now that we know that Grant's airplane map contains a right triangle, let's think more about right triangles in general. I've done three different right triangles for you right here. And below each one, I've written one equation or in these two cases two equations that relate the lengths of their sides to each other. For each triangle, I've labeled a as the shortest side, b as the second shortest side, and c as the longest side link. Which equation or set of equations down here will hold true for all right triangles?
The answer is this equation right here, which goes with this triangle, a squared plus b squared equals c squared. These other two right triangles are a special kinds of right triangles and the equations right here that show the proporions between their side lengths are particular to them individually and they don't apply to all right triangles. But, now we do have a super important piece of information that we can use, not just to talk about these three right triangles, but also the one in Grant's situation and any other ones moving forward. This equation, a squared plus b squared equals c squared, relating the side lengths of any right triangle together is called the Pythagorean theorem. Julie mentioned this way back in the very first lesson of this course. So, no big deal if you didn't remember it or its name. We'll keep using it in the future, though.
We just talked about how the Pythagorean theoem says that if you have a right triangle, and you square the shortest side, and add to that the square of the second shortest side, then what you get as the sum is the square of the longest side of the triangle. The longest side of a right triangle is also known as the hypoteneuse, that's a word you may have heard before. What I would like you to tell me is how long the hypoteneus of this triangle is. Should this question mark be replaced by 225, 15, 12, 9 root 3, 9 root 2, or 9?
The correct answer is 15. We know, from the Pythagorean Theorem, that the sum of the squares of either of the shorter sides of the triangle is equal the square of the hypotenuse. So, in the case of this triangle, that means that we have 9 squared plus 12 squared equals something squared. We want to find out what c is. Simplifying the left side of our equation gives us 225 equals c squared. Now it's really easy at this point to think, oh my gosh, I found our answer. But remember, we're not looking for c squared, we're looking for just c on its own. We just want the length of this side, not its square. So how do we handle this? How do I get rid of this exponent of 2? Well, we need to talk a little bit more about exponents, so we can figure out how to deal with this. What we need to do is to take the square root of both sides. You've probably worked with square roots before and the power of the square root is just transform, in this case, c squared to just c. It also changes 225 into 15. We'll elaborate on square roots in just a minute.
Thinking about the last problem you just did and your knowledge of exponents, what do you think a square root gives you? We have some number n and I take the square root of it, what is the new number that we get?
The correct choice is this third one. For any number n, the square root of n is equivalent to some number that when multiplied by itself, gives us n. I know that this is a lot of words, so let's just break that down. Squaring a number or taking it to the second power, we know means, multiplying it by itself. So, if we have 3 squared, that's just two threes multiplied together. But I know that if I take the square root of 3 squared, then I can get just the 3 back. So, taking a square root can undo this repeated multiplication. The square root of any number then like n, for example, tells us what we need to multiply by itself to get n.
Since we know there's some connection between square roots and exponents, let's take a minute to make sure you're still comfortable with exponents themselves. Please tell me what y to the 7th times y to the negative 3rd is.
y to the 7th times y to the negative 3rd is just y to the 4th power. Remember that shortcut we learned before, for any numbers n, a, and b, n to the a times n to the b is just equal to n to the a plus b power. So here, all we need to do is add our exponents together, 7 minus 3 is equal to 4.
We've been thinking about squaring things a lot with those right triangles. So what if we add x cubed squared? What is that equal to?
The thing that we want to square here, or multiply by itself, is x cubed. So, all we have here is just x cubed times x cubed. And what is that? That's just x to the 6 power.
So I'm noticing something pretty interesting in here. A pattern that relates these numbers together. What if I want to know this time what y squared to the fourth power is?
Y squared to the 4th is just 4y squareds all multiplied together. And that, using our shortcut is just y to the 2 plus 2 plus 2 power which is just y to the
Here are the problems you did in the last couple of quizzes. Did you notice a pattern at all while you were doing them? If I have a general statement, x to the a all to the b power, how could I express this with a single exponent? In other words, what happens if I want to get rid of the parentheses around the x to the a here?
The answer is that the exponent here should be a times b or just ab. This expression over here says multiply x to the a times itself b number of times, but we also know that x to the power of a tells us to take x, and multiply it by itself a number of times. If we wrote each of these factors of x to the a as a bunch of x's multiplied together, we would just get a giant string of x's all multiplied together. Since we're taking b groups of a things multiplied together, the total number of x's we have is a times b.
We said earlier that the square root of the number n is just a number that when multiplied by itself, gives us n. Let's pick a real number to make this more concrete for ourselves, though. Right now, I am feeling 3. 3 is the number that I want to use right now. But what number times itself gives us 3? Well, we just said that was the whole point of square roots. So clearly, the square root of 3 times the square root of 3 equals 3. But this is interesting. The square root of this mean for square roots in terms of exponents? What if I wanted to rewrite this as 3 to some power, all of that squared, equals 3. What number belongs in this box?
We need to put a 1 half here. Remember that the guideline we just talked about said that if we have some number x to the power a and all of that taken to the power of b, that's just equal to x to the power of a times b. We can't forget that there's an invisible 1 as the exponent of the number 3. So, the power that belongs right here must be whatever we need to multiply by 2 to get 1, and that number is 1 half. This also makes sense in the context of another rule that you know, x to the a times x to the b is equal to x to the a plus b. 3 to the 1 half times 3 to the 1 half is just equal to 3 to the 1 half plus 1 half or 3 to the
We just learned that the square root of some number n can be written in exponent form as n to the 1 half power. That's great, that's super useful. What if instead though, I have something y to the 5th, and I want to do something to this using exponents that I end up with just y by itself, what exponent belongs in this box?
The number that we need to multiply five by to give us one is just 1 5th. So that's the exponent that belongs right here.
What number squared equals 64? As much as it might surprise you, there are actually two different numbers that satisfy this. I'd like you to tell me both of them please.
Well, we know that 8 times 8 equals 64. So 8 is definitely one of the answers, but, there's one more. Negative 8 times negative 8 is also equal to 64. If we want to expound upon why this is true, we can write out that each negative 8 is equal to negative 1 times 8. Use the commutative property to rearrange all those factors, and since negative 1 times negative 1 equals positive 1, we still end up with positive 64, so our other answer is negative 8.
So this tells us that there are actually two square roots for any number n. There's the positive version, which of course squared gives us n but there's also the negative version. Which, also 1 squared, gives us n. You will notice, however, that although there's a negative sign written outside the negative square root. There's not a positive sign written outside the positive version. The positive version of the square root is implied to equal a positive number. For example, if we wrote the square root of 64, with no sign symbol out in front, this would be implied to equal positive 8. If we wanted to show that negative 8 times itself equals 64 then we'll write the square root sign at 64 but we'd also need to put a negative sign out in front of it. The square root of a number without the negative sign in front is called the principal square root of that number. In this case, n. So, 8 is the principal square root of 64.
So considering this, what is the principal square root of 81? Write it in this box please.
We know that there are two numbers when multiplied by themselves that equal 81. We know that 9 times 9 equals 81 and we also know that negative 9 times negative though. We get 9, positive 9, if we just write the square root of 81 and we get negative 9 if we write negative square root 81. The principal square root is a square root that doesn't have a negative sign out in front of it, the positive square root, so the answer is 9. The principle square root of any number is a positive number, that times itself, equals that number.
We've already seen that square roots can be expressed equivalently with either radical signs or with exponents of one half, but as a matter of fact, all exponent of 1 over n, where n is a natural number, can be expressed with a radical sign But we need to differentiate between this radical sign and this radical sign, or all the different radical signs we would use as this number n changes. To do this we put a little number next to the radical sign. To denote x to the 1/3 or the cube root of x, for example, you put a little 3 by the radical sign. The question I have for you is, if we want to express the nth root of x, what symbol belongs next to the radical sign?
We need to put a little n next to our radical sign to show the nth root of x or x to the 1 over n power. So if we write the 5th root of x for example, that's just equal to x to the 1 5th, and similarly, for the 8th root of x. Square roots then, are the one xception to our rule. It's like there's an invisible two written next to the radical sign here, but according to convention, we just don't write any number for square roots.
You've been using MathQuill for the past few lessons, and hopefully you've gotten used to entering exponents and fractions using it, making your equations and expressions look super pretty. But now that we're starting to use radicals, we need to use MathQuill in a new way. I already showed you earlier how to enter square roots in MathQuill, you type backslash sqrt space, and then whatever goes inside. The MathQuill also lets us write nthroots. So, it allows us to type radical signs and then type whatever number we want for the index, and then, of course, enter whatever we want underneath the radical. To write a radical sign when you need to write a number you just write backslash nthroot space, and then whatever number you want to go in this spot up here. To get out of typing in the index spot, to typing underneath the radical sign again, you need to type a right arrow key. So, try out these first two where you can use some instructions on what to type. And then, for your two challenge problems, I'd like you to experiment combining the different functionalities in MathQuill that we've been using with this new set of tools. So, we have, for example, in this term, exponents combined with radicals and fractions all in one. So, if you can do this, you're an official MathQuill pro. Just experiment, just have fun to see how incredibly powerful this tool is.
I'd already given you instructions for how to type these two right here, the square root of 2 and the cubed root of x. But then, as you can see for these two slightly more difficult ones, when you write out all the keys you have to type in MathQuill, it looks super complicated. But hopefully, if you found it's a little more intuitive when you're actually typing it, especially because you've already had practice using MathQuill before. Please remember that when you type parentheses in MathQuill, it actually fills in the second parentheses when you type the first one, so you don't have to retype at the end.
What if we have the 4th root of x cubed? What's a way to write that with only one exponent?
Our answer is 3 4th. We can rewrite what's inside the parenthesis here as x to the 1 4th power, that's what we want to cubed. From here, we can use one of those lovely shortcuts we talked about earlier. This is the same as x to the 3 times 1 over 4, which is the same as x to the 3 4th.
How can we rewrite 5xy cubed, all of that squared? I want you to rewrite this without any parentheses involved.
The thing that we're squaring here is everything inside the parentheses. So, we just want to multiply 5xy cubed by itself. If we expand this multiplication and then simplify, we end up with 25x squared y to the 6th. What we can see is that our end result here, takes each factor from the original quantity inside parentheses and squares it. So, 5 squared is 25, x squared is x squared, y cubed squared is y to the 6th. So, you can think of this as the exponent distributing to every single factor that's multiplied together inside the term.
Now, it is time for a, get ready, a challenge problem, yay. I'm going to write some stars for you. Anyway, please tell me what the cube root of negative 8x to the 6th power is.
Let's start by changing this radical sign to a power. Cube roots give us powers of 1 over 3. Notice that I put the parentheses around the entire quantity, negative 8x to the 6th since the radical sign extended over all of those factors. Now, all we need to do is what we did in the last quiz, distribute this exponent to both of the factors in the term. That gives us negative 8 to the 1 itself 3 times to get negative 8 is negative 2. So, that's our coefficient. And the cube root of x to the 6th is just x squared. So, our final answer is negative 2x squared.
Let's think back way, way back to right triangles. Let's say that I have a right triangle whose two shorter legs are both equal to the square root of 2 times x. What then, is the length of the hypotenuse of this triangle or the length of this longest side? Be super careful about which factors are under the radical sign.
To make our Algebra a little bit easier, I'm going to call the length of the hypotenuse l for just a moment. We're just going to use this other variable to describe it. We know from the Pythagorean Theorem, that if I square each of these sides and add them together, I'll get l squared. Then, we just need to simplify this left side of the equation. I could distribute the exponent of 2 to all of the factors and then add like terms. Since we want to get l by itself, we need to take the square root of both sides. Or in other words, I need to take both sides to the 1 half power. On the right side I just get l and on the left side, I get 2x. So, our final answer is, that the length of the hypotenuse equals 2x.
Now, where were we with Grant? Oh, that's right. He was on an airplane, going from his hometown to this wonderful, wonderful city called Udacity. We reduced that map to a right triangle, just like this one. Now, can you tell me how long the flight path from his hometown to Udacity is? Please round to the nearest whole number.
For the moment I'm going to call the length of our flight path just the letter d. That will make dealing with it in equations much simpler. The Pythagorean theorem says that because this is a right triangle, we can say that 200 squared plus 700 squared equals d squared. Simplifying that gives us 530,000 equals d squared. In order to get d by itself, what do we need to do? Just take the square root of both sides. Taking the square root of d squared, just leaves us with d. The square root of 530,000 gives us a number with a ton of decimal places, but when we round it, we get 728, which, since we're rounding, I will say is about equal to d. Since our flight is really kind of approximate any way, we can add to it that this length from Grant's hometown to Udacity is about 728 miles.
What kind of number was the exact answer to the last problem, the non-rounded one?
This is an irrational number. And because it's an irrational number, just like all the other numbers we've talked about, it's also a real number. We know this because the square root of 530,000 has an infinitely long decimal with no repeating pattern in it. In fact, any root of some integer, m, is either an integer or an irrational number. That's just kind of a fun fact for you to know, and something useful for you to check answers later on.
Over the course this lesson, we discover a super powerful tool because of Grant's airplane situation. We can now find the distance between any two points on a graph. And, I don't just mean points that we can count squares on graph paper to find the distance between. I mean, any two points on a coordinate plane. Think about how you used triangles, right triangles in particular, to find the distance from Grant's hometown to the city Udacity. Can you do something similar to find the distances between these points on this graph? To make your life easier, I only want you to find the distance from A to B, from B to C, from C to D and from D to A. Please write your answers in exact form. In other words, I don't want you to round. Instead, I would like you to write them in whatever form you need to write them to represent the actual exact number that each length is equal to. To help you do this, I'd like to remind you that if you want to enter a square root sign, using MathQuill, you need to type the keys back \sqrt, and then the space bar. Then, you can use your arrow keys to move your cursor around.
What we need to see here in the graph is a bunch of triangles. So, if we let each side length, or each of these distances that we're looking for, be the hypotenuse of a right triangle, then we can as we've done so many times before, use the Pythagorean theorem to find the length of that side. Conveniently, since we're using graph paper, made up of these little tiny squares, we have right angles all over the place. That made drawing these right triangles super simple. If you look at the opposite ends of one of these side lengths, we can just draw lines along the lines that are already given to us by the graph paper. And the intersect to help us form the right triangle. Then, we just need to figure out the length of each of these legs for each of these triangles. Let's look at leg AB as an example for how to deal with the other three triangles on here. We need to find out the length of either of the two shorter legs in order to find out the hypotenuse. So, we could just count one, two, three, four, five, six, seven. Seven is the length of this side. We'll do the same thing for this side. This is five. In looking at this modified version of the Pythagorean theorem, we can just say that the length of AB, this hypotenuse, is the square root of 7 squared plus 5 squared. Simplifying that gives us the square root of 74. Once we find the lengths of the legs of all of the other right triangles, calculating their hypotenuses is also easy. And so, at long last, here are our final answers. The length of AB is the square root of 74, BC, square root of 180, CD is the square root of 40, and DA is the square root of 130.
We just calculated that the exact length of segment BC was the square root of remind you that 180 is actually equal to 36 times 5? So, the square root of 180 is equal to the square root of the quantity 36 times 5. We also know, though, that 36 is the same as 6 squared. Taking this into account, can you think of a different way to write the length of BC, so that there's one number under a radical sign and another number multiplied by that that's not under the radical anymore?
The answer is 6 root 5, or 6 times the square root of 5. There are however a couple of steps between the last step I showed you and this answer, so let's talk about them. We know that square roots are the same as exponents of 1 half, So you can rewrite this as 6 squared times 5 all to the 1 half power. You learned earlier though about how exponents distribute to factors inside a term. That means this is also equal to 6 squared to the 1 half times 5 to the 1 half, which of course we can return to radical notation. The square root of 6 squared is just 6, and the square root of 5 is an irrational number. ,
When we rewrite things like the square root of 180, instead as 6 times the square root of 5. We say that we've written this radical expression in simplest form. What this means is that the number that's still under the radical sign, or squared, where n here is some rational number. Since the square root of 40 was one of the other side links in our quadrilateral earlier, I would like you to try writing it in its simplest form.
We know that 40 is equal to 4 times 10. So, we can replace the number under the radical with that expression instead. We know that this radical sign here represents a 1 half power outside this entire expression. So, the radical sign itself should just distribute to the factors inside this term. That means, we can write that this is equal to the square root of 4 times the square root of just equal to 5 times 2. These are the prime factors of 10. Neither of these is a square. So , that means that we can just leave the square root of 10 as the square root of 10. This is the simplest form of the square root of 40.
Now that you know how to write expressions with square roots in simplest radical form, can you do it with an expression that has a cube root in it instead?
The answer is 3 cube root 2. Let's see how to get that. 54 is just equal to 27 times 2. We can rewrite this as an expression with exponents and then distribute the exponent to the factors. 27 is equal to 3 to the 3rd power. So, this reduces to 3 times 2 to the 1 3rd power or just 3 cube root 2.
Thinking about how you rewrote the cube root expression in the last quiz in simplest radical form, can you come up with a general rule for an expression with any sort of root in it? If we have a radical expression a times the nthroot of b, when will it be written in simplest radical form?
a times the nth root of b is written in simplest radical form, as long as b doesn't have any factors that can be written as powers greater than or equal to n, the number that tells us what kind of root this is. The reason for this is, if we have something like the 4th root of 7 to the 6 power, we can rewrite that as the 4th root of 7 to the 4th times 7 to the 2nd. This first factor will just reduce as a result of the radical to be 7 on its own, and all that will be left under the radical sign, or rather the 4th root sign will be 7 squared. And of course, we can also rewrite this as 7 times the 4th root of 49.